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Theorem fvopab4ndm 6620
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
Hypothesis
Ref Expression
fvopab4ndm.1 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
Assertion
Ref Expression
fvopab4ndm 𝐵𝐴 → (𝐹𝐵) = ∅)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvopab4ndm
StepHypRef Expression
1 fvopab4ndm.1 . . . . . 6 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
21dmeqi 5619 . . . . 5 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
3 dmopabss 5631 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
42, 3eqsstri 3884 . . . 4 dom 𝐹𝐴
54sseli 3847 . . 3 (𝐵 ∈ dom 𝐹𝐵𝐴)
65con3i 152 . 2 𝐵𝐴 → ¬ 𝐵 ∈ dom 𝐹)
7 ndmfv 6526 . 2 𝐵 ∈ dom 𝐹 → (𝐹𝐵) = ∅)
86, 7syl 17 1 𝐵𝐴 → (𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1508  wcel 2051  c0 4172  {copab 4987  dom cdm 5403  cfv 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-dm 5413  df-iota 6149  df-fv 6193
This theorem is referenced by:  fvmptndm  6621
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